I'm trying to understand what is the point of the local volatility model in practice. Rather than asking a question I will explain what is what for me hoping someone will spot where I'm wrong:

The point of using another model than Black-Scholes is "roughly" to capture more market informations in a better way obviously, in order to price derivatives mostly (even exclusively).

Now the main subject of "how to price derivatives" can be rephrase as "how to interpolate the implied volatility", am I right?

But since that, in order to calibrate the local volatility model, you need to interpolate the implied volatility first, how does using this model can make sence in practice?


Some points below as food for thought:

  • Suppose you possess an implied volatility surface over a continuous strike cross time to expiry domain (how to get there from the discrete market specification is another question). Further assume that you have to price a path-dependent option, e.g. a Barrier or an Asian. If you are using Black-Scholes, what implied volatility number are you going to plug. For the barrier option: at what 'strike' of the IVS will you look? The barrier level or the strike of the option? For an Asian option, at which 'time' of the IVS will you look: the maturity date or any other asianing date? The local volatility model gives a simple answer to that question: once you have stripped the Dupire volatilities from the IVS you always use that surface and that's it.
  • More philosophically, a pricing model should never be reduced to a black box returning a price. Rather a model tells you what (self-financing) dynamic strategy you can set up to replicate an option's value using the relevant quantity of marketed securities of your model economy. The model is a fair one if the replication error is zero in expectation between each rebalancing period. This typically gives rise to an equation involving: (i) the Greeks of the instrument to be priced; the market dynamics (realised variations of the hedge instruments' prices) and the model dynamics (payoff-independent quadratic variations of the hedge instruments' prices). There are then two key points to consider. The [Statics] of a model: how well it can capture the current picture of the vanilla market. The [Dynamics] of a model: how plausible are its break-even levels.
  • [Statics] Dupire volatilities are constructed so that all vanilla option prices are perfectly matched. Typically, this means that all instruments that can be expressed as a linear combination of vanillas will be priced and hedged consistently. So LV is perfect in that case.
  • [Dynamics] Consider second generation exotics like forward start options. The information required to price these options is not encoded in the vanilla market (conditional risk-neutral distributions vs. unconditional risk-neutral distributions). You will then depend on the assumptions embedded in your pricing model. Typically, LV is known to generate unrealistic implied volatility dynamics (+ the break-even levels will depend on the current market conditions which is not practical from a risk-management perspective. That's what you get from using a non-parametric model).

As always for these topics, I recommend reading Lorenzo Bergomi's excellent book "Stochastic Volatility Modeling". Sample chapters are available here.

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    $\begingroup$ A [simple but] key point you make IMO is that Dupire is intended to be applied to non-vanilla derivatives. For vanilla the observed IV's (perhaps slighlty smoothed or cleaned up) can simply be plugged into the BS formula to give the price. Dupire aims to provide non-vanilla prices consistent with observed vanilla IVs. $\endgroup$
    – Alex C
    Sep 14 '18 at 9:22
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    $\begingroup$ Agreed @Alex C. I guess it depends on what one means by the term 'vanilla'. For instance BS can be used to price binary options. But the price returned will be irrelevant since one cannot account for skew in BS (except if you use an ad hoc correction, or the call spread approximation which then requires 2 pricings). $\endgroup$
    – Quantuple
    Sep 14 '18 at 9:36

The benefit of Dupire formula is that you can find a local volatility function from the market price of vanilla options, then you can use these local volatilities (by constructing a surface) to value exotic options. Also, this single (theoretically) unique volatility function will value all the vanilla options in line with their market prices rather than having to use the inconsistent Black Scholes model with different volatilities for each vanilla option.


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